Asymptotically Isometric Copies of c0 and l1 in Bochner-Spaces

Let X be a Banach space and μ be a finite measure space. It is shown that if 1 ≤ p < ∞ resp 1 < p < ∞ , the Bochner space Lp μ X contains asymptotically isometric copies of c0 resp l1 if and only if X does.